Simultaneously Colouring the Edges and Faces of Plane Graphs

In a simultaneous colouring of the edges and faces of a plane graph we colour edges and faces so that every two adjacent or incident pair of them receive different colours. In this paper we prove a conjecture of Melʹnikov which states that for this colouring every plane graph can be coloured withΔ+3 colours, whereΔis the maximum degree of a vertex in the graph.